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Quadratic Equations
An equation whose highest power is 2 is an quadratic equation . The graph of an quadratic is always a parabola . Solution of a Quadratic Equation A Quadratic Equation cannot have more than two roots . Completing the Square Method Factorization Method Formula Method x =±√(b.2.-4ac) / 2a Substitution Method Form aZ +b/Z + c Form (x + a)(x + b)(x + c)(x + d) + k = 0 Step 1 :De-factorize (2 at a time) Step 2 : Substitute [ if a + b = c+ d ] Step 3 : Solve and Re-substitute Complex & Irrational Quadratic Equations If any coefficient is a non-real complex and p + iq is a complex root of equation , then p - iq need not b the other root . If any coefficient is an irrational number and p + √q is a complex root of equation , then p - √q need not b the other root . The above holds true for even higher degree polynomials . Relation between Roots and Coefficients α + β = - b/a α β = c/a Nature of Roots b2 - 4ac > 0 Roots are Real and Unequal b2 - 4ac = 0 Roots are Real 'and '''Equal . The equation is a perfect square ' b2 - 4ac < 0 Roots are '''Complex b2 - 4ac is not a perfect square Roots are Real 'and '''Irrational ' Sum and Product of Roots α + β = - b/a αβ = c/a Formation of Quadratic Equation x2 - (Sum of Roots)x + (Product of Roots) = 0 Sign of Roots {Based on laws of inquations} 'Positive , ' if both sum and product of roots are positive . 'Negative , ' if both sum and product of roots are negative . 'Opposite Signs , ' if product of roots is negative . 'Equal Magnitude and Opposite Signs , ' if Sum of Roots is 0 . Locations of Real Roots Graph Parabolic Graph b2 = 4ca y2 = 4ax Vieta's Formulas Quadratic Inequalities Quadratic Inequalities are solved in the same way as quadratic equations (Interpretation is important) . Symmetric Functions of Roots Quadratic & Higher Order Polynomials A polynomial of n order , has n roots . Note the similarities in quadratic equations and higher order polynomials . Consider α β γ to be roots of third order polynomial ax3 + bx2 + cx + d ; then α+β+γ = -b/a αβ + βγ + αγ = c/a αβγ = -d / a Consider α β γ δ to be the roots of a fourth order polynomial ax4 + bx3 + cx2 + dx + e ; then α+β+γ+δ = -b/a αβ + βγ + γδ + αγ + αδ + βδ = c/a αβγ+ βγδ + αγδ + αβδ = -d/a αβγδ = e/a Common Roots in two Quadratic Equations α is a common root in a1x2+b1x+c1 = 0 and a1x2+b1x+c1 = 0; thus α '''satisfies both equations . If α β are the roots of first equation ; α γ are the roots of second equation ; then Sum and Products of roots of both equations can be equated due to the common root α . (This helps in solving such equations) Category:Mathematics